question_answer

Mass of a planet is\[1/{{9}^{th}}\]of the mass of the Earth. Radius of the planet is 1/2 of that of Earth. If a man weighs 9 N on Earth, his weight on the planet is (a) 9 N                                   (b) 244 N   (c) 488 N                              (d) zero

Answer:

               

Earth	Planet
\[{{M}_{1}}=M\]	\[{{M}_{2}}=M/9\]
\[{{R}_{1}}=R\]	\[{{R}_{2}}=R/2\]
\[{{W}_{1}}=9N\]	\[{{W}_{2}}=?\]

We know that, \[W=mg\] \[\Rightarrow \] \[W\propto g\] (\[\because \]m is constant) \[\Rightarrow \] \[\frac{{{W}_{2}}}{{{W}_{1}}}=\frac{{{g}_{2}}}{{{g}_{1}}}\] How to get \[\frac{{{g}_{2}}}{{{g}_{1}}}?\] We know that,\[g=\frac{GM}{{{R}^{2}}}\]\[\Rightarrow \] \[g\propto \frac{M}{{{R}^{2}}}\] (\[\because \]G is constant) \[\Rightarrow \]\[\frac{{{g}_{2}}}{{{g}_{1}}}=\frac{{{M}_{2}}}{{{M}_{1}}}\times \frac{R_{1}^{2}}{R_{2}^{2}}=\frac{M/9}{M}\times \frac{{{(R)}^{2}}}{{{(R/2)}^{2}}}=\frac{4}{9}\] \[\therefore \]\[\frac{{{W}_{2}}}{{{W}_{1}}}=\frac{{{g}_{2}}}{{{g}_{1}}}=\frac{4}{9}\] \[\therefore \]\[{{W}_{2}}=\frac{4}{9}\times 9=4N\]